Edwards's Solution of Pendulum Oscillation

The masterful derivation by Harold Edwards [1] finally brings the vision of Abel to the wider audience it deserves. In addition to its elegance, the article provides a constructive approach for improving computations along elliptic curves. The new and simple addition rules have been widely appreciated, although less so for the ingenious function introduced in [1, Section 15]. As with the much earlier Weierstrass function, the Edwards function determines time-dependent solutions for a range of interesting Hamiltonian systems [2]. This Demonstration shows three interrelated examples, including one that describes the oscillation of a plane pendulum. The Edwards function is truly an amazing and beautiful, doubly periodic, meromorphic function!

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DETAILS

Beginning with Edwards' normal form
,
apply the transformation
to derive the Hamiltonian form:
.
Each level set is a quartic elliptic curve with period energy function
.
This Hamiltonian double covers another,
.
The contours of with are hyperelliptic curves where the Jacobi elliptic functions naturally arise in the time-parametrization problem. This surface looks astoundingly similar to a model for the plane pendulum,
.
In fact, the second and third Hamiltonians are related by a canonical transformation, which leads to the exact solution for the equations of motion of a plane pendulum.
This can be readily applied, since we need only a first approximation of the Edwards doubly periodic, meromorphic function,
.
One issue remains with this simple form, the calculation of . As with cubic anharmonic oscillation, the Picard–Fuchs equation is just a hypergeometric differential equation, so that
.
Using this function, we can immediately determine time-dependent solutions of motion along .
The other two Hamiltonian surfaces have equations of motion that depend on the Jacobi functions and ,
,
,
,
,
for and respectively. The Jacobi functions have the same pole-and-zero structure as , so it is possible to represent both and in terms of . Restricting the domain to , the simplest instance suffices,
.
However, the function is not really designed for this purpose. We need to do some more hacking before this identity works. The function that satisfies
provides the necessary convergence acceleration. We can then propose the solution
.
Proof that satisfies the requisite constraint reduces to routine evaluation of a binomial identity [3, 4],
.
Zeilberger's algorithm accepts the right-hand side as input, so the identity can be proven automatically. The RISC Mathematica implementation returns the appropriate hypergeometric parameters [4]. Finally, if is the parameter of a Jacobi function, the corresponding period ratio for is .
Very optimal expressions exist for calculating with the Jacobi functions . The built-in Mathematica functions JacobiCN and JacobiSN can do this. However, in our own timing tests, it appears that evaluation of the Edwards function is sometimes faster by about a factor of 10.
References
[1] H. M. Edwards, "A Normal Form for Elliptic Curves," Bulletin of the American Mathematical Society, 44, 2007 pp. 393–422. doi:10.1090/S0273-0979-07-01153-6.
[2] B. Klee. "Weierstrass Solution to Cubic Anharmonic Oscillation" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/WeierstrassSolutionOfCubicAnharmonicOscillation.
[3] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences. "A002057." oeis.org/A002057.
[4] N. J. A. Sloane and P. D. Hanna. The On-Line Encyclopedia of Integer Sequences. "A119245." oeis.org/A119245.
[5] P. Paule and M. Schorn, "A Mathematica Version of Zeilberger’s Algorithm for Proving Binomial Coefficient Identities," Journal of Symbolic Computation, 20(5–6), 1995 pp. 673–698. doi:10.1006/jsco.1995.1071.
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